Detectives Nemorin and Andrews Anti-Gun Trafficking Act of 2013 - Amends the federal criminal code to: (1) impose a fine and/or prison term of up to 20 years for the sale, transfer, or barter of a stolen firearm or a firearm with an altered serial number or for the sale of such firearms to persons prohibited from possessing them; (2) impose an additional five-year prison term for possession of a stolen or altered firearm during the commission of a felony; (3) require licensed firearms importers, manufacturers, or dealers to contact the national instant stolen gun check system before receiving firearms from any unlicensed person; and (4) increase criminal penalties for the use of a stolen or altered firearm during the commission of a crime of violence or drug trafficking crime.

Amends the 21st Century Department of Justice Appropriations Authorization Act to: (1) include illegal gun trafficking in Department of Justice (DOJ) programs to combat violations of federal firearms laws, and (2) authorize appropriations for FY2013 to hire additional assistant U.S. attorneys.

Directs the Federal Bureau of Investigation (FBI) to make the National Crime Information Center Gun File available to the Bureau of Alcohol, Tobacco, Firearms, and Explosives (ATF) to complete crime gun traces. Requires the ATF to: (1) notify law enforcement agencies of information relating to stolen firearms, and (2) return stolen firearms to persons who reported them as stolen.

Requires the Attorney General to establish a national instant stolen gun check system.